IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
1
State Variable-Based Transient Analysis Using
Convolution
Carlos E. Christoffersen, Student Member, IEEE, Mete Ozkar, Student Member, IEEE,
Michael B. Steer Fellow, IEEE, Michael G. Case, Member, IEEE, and Mark Rodwell, Member, IEEE
Abstract—
A state variable-based approach to the impulse response
and convolution analysis of distributed microwave circuits
is developed. The state-variable approach minimizes computation time and memory requirements. It allows the use
of parameterized nonlinear device models thus improving
robustness. Soliton generation on a nonlinear transmission
line is considered as an example.
stant time steps can be used further improving the robustness of the convolution.
A nonlinear transmission line is used here and it is regarded by many in the field as an extreme test of the performance of transient and steady-state simulators. More
generally, the work is directed at the transient simulation
of distributed systems with tightly coupled circuit-field interactions.
I. Introduction
T
RANSIENT analysis of distributed microwave circuits
is complicated by the inability of frequency independent primitives (such as resistors, inductors and capacitors)
to model distributed circuits. More generally, the linear
part of a microwave circuit is described in the frequency
domain by network parameters especially where numerical field analysis is used to model a spatially distributed
structure. Inverse Fourier transformation of these network
parameters yields the impulse response of the linear circuit.
This has been used with convolution to achieve transient
analysis of distributed circuits [1], [2], [3], [4].
The major drawback of convolution analysis has been
the large time and memory requirements that result from
recording and convolving the nodal voltages of the nonlinear elements. The situation worsens because of convergence
considerations which require small time steps. This paper
develops a convolution-based transient analysis which uses
state variables instead of node voltages to capture the nonlinear response. This has two desirable effects. First, the
number of state variables required is less than the number
of nodes of the nonlinear elements and so fewer quantities
need to be recorded and convolved. Second, parameterized
device models can be used as the nonlinearity is not restricted to the form of current as a nonlinear function of
voltage. Parameterized device models result in stable transient analysis for larger time steps and so less discretized
time history is required. Also with improved stability con-
This work was supported by the Defense Advanced Research
Projects Agency (DARPA) through the MAFET Thrust III program
as DARPA agreement number DAAL01-96-K-3619.
C. E. Christoffersen and M. Ozkar are with the Department of Electrical and Computer Engineering, North Carolina State University,
Raleigh, NC 27695-7911, U.S.A., (e-mail: c.christoffersen@ieee.org,
mete ozkar@ieee.org).
M. B. Steer was with NCSU. He is now with the Institute of Microwaves and Photonics, School of Electronic and Electrical Engineering, University of Leeds, Leeds, United Kingdom LS2 9JT, (e-mail:
m.b.steer@ieee.org).
M. G. Case was and M. Rodwell is with the department of Electrical
and Computer Engineering, University of California, Santa Barbara,
CA 93106, U.S.A.
II. Background
Distributed linear microwave circuits are described by
frequency-dependent network parameters and only a few
methods are available to accommodate these circuits in
transient simulation. These methods include Impulse
Response and Convolution (IRC), Asymptotic Waveform
Evaluation (AWE), and Laplace Inversion.
A. Impulse Response and Convolution
One of the first implementations of IRC to distributed
microwave circuits was by Djordjevic and Sarkar [5] in
1987. Since that time there have been several efforts to
reduce the significant aliasing errors that result from the
Inverse Fast Fourier Transform (I-FFT) operation.
Minimization of aliasing in the I-FFT requires that the
imaginary part of the frequency response be zero at the
maximum frequency. Low pass filtering has been used to
achieve this [6]. The introduction of a small time delay also
achieves this result with presumably less distortion [4]. Insertion of an augmentation network in the linear network at
the interface with the nonlinear network achieves the same
result for special types of circuits [1]. The effect of the augmentation network is compensated in the nonlinear iteration scheme. It is also important to limit the length of the
impulse response to reduce memory requirements and the
resistive augmentation achieves this result [1]. Augmentation is also effectively achieved using s parameters [7],
where the reference impedance effectively dampens multiple reflections.
Distributed networks are characterized partly by reflections so that an impulse response tends to have regions
of low value between regions of rapid change. In this
case thresholding greatly reduces the number of impulse
response discretizations that need to be retained [1]. In
high speed digital interconnect circuitry this can reduce
the number of significant impulse responses by a factor of
10 to 100, depending on the desired accuracy [1].
Convolution as generally practiced uses a rectangular in-
2
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
tegration scheme (essentially an impulse response is treated
as being constant in a time step interval). P. Stenius et
al. [8] developed a trapezoidal form of the convolution integration which could possibly have superior convergence
properties than the previous block integration.
B. Asymptotic Waveform Evaluation
The frequency dependent network parameters can be
modeled using frequency independent primitives (resistors,
inductors and capacitors) if a rational polynomial transfer
function is fitted to the network parameters. In practice,
this procedure results in an impossibly large circuit. The
AWE method addresses this problem by reducing the dimension of the rational polynomial while minimizing distortion [9], [11]. This method works well for interconnects in digital systems and lower frequency microwave circuits [12], [13]. However, higher frequency circuits can only
be modeled approximately using AWE due to the infinite
number of poles and zeros of such a circuit.
Most of the AWE methods use the Padé approximation
and this and other approximations used can have stability
problems [9].
Recent work has extended this technique to more general
distributed circuits [10].
C. Numerical Inversion of Laplace Transform Technique
This technique does not have aliasing problems since it
does not assume that the function is periodic. The inverse transform exists for both periodic and non-periodic
functions. There is no causality problem for double sided
Laplace Transforms, either. Unlike FFT methods, the desired part of the response can be achieved without doing
tedious and unnecessary calculations for the other parts
of the response. Laplace techniques suffer from the series approximations and the nonlinear iterations involved.
The advantages and the limitations of the Inverse Laplace
Methods are discussed in detail in [14].
III. Formulation of the Transient Analysis
As it is now conventional for distributed microwave circuits, the circuit is partitioned into linear and nonlinear
subcircuits [15]. The core of the method presented here
is that state variables are used to describe the nonlinear
dependence in the nonlinear subcircuit. This enables more
flexibility in writing the nonlinear element models so that
better convergence might be achieved [20].
The linear subcircuit is characterized in the frequency
domain and the other part, which includes the nonlinear
devices, is treated in the time domain. From [16], the vector of voltages VL in the frequency domain calculated at
the interface to the linear network is
VL (X, f) = SSV (f) + MSV (f) INL (X, f)
(1)
where X is the state variable vector, MSV is the state variable impedance matrix and INL is the vector of currents
flowing into the linear network at the interface of the linear/nonlinear network. Since the independent sources are
more easily handled in the time domain for this kind of
analysis, the source vector Ssv (f) can be assumed to be
zero and the sources are considered together with the nonlinear devices. Expanding the matrix multiplication, each
element of the voltage vector VL (X, f) can be written as:
VL i =
nS
X
Mi,j (f) INL,j
(2)
j=1
Rewriting one term of this equation in the time domain
and replacing the multiplications with the convolution operation leads to
vLi,j (t) = mi,j (t) ∗ iNLj (t)
(3)
Now, expanding the convolution operation we get:
Z t
vL i,j (t) =
mi,j (τ ) iNL j (t − τ ) dτ
(4)
−∞
where the system is assumed to be causal, iNL j (t) = 0 for
t ≤ 0. Numerical evaluation requires discretization as follows. First, each element of the I-FFT of MSV has a finite
number of components, denoted NT . Using the trapezoidal
integration rule [8] and using mi,j obtained from the I-FFT,
we obtain Eq. (5). For nt < NT the last term is zero since
ij (0) = 0. For nt ≥ NT the last term is also zero since
mi,j (nt ) = 0.
vLi,j (X̂, nt ) =
 m (0) i (X̂,n )
i,j
NL
t


2

P
n
−1
 +
t


nτ =1 mi,j (nτ ) ij (nt − nτ ),


if nt < NT
mi,j (0) iNL j (X̂,nt )



PNT 2



 + nτ =1 mi,j (nτ ) ij (nt − nτ ),

if nt ≥ NT
(5)
Note that in all but one of the convolution sum terms,
iNL = iL = i, so most of the convolution sum can be
performed before beginning the iterations to solve the nonlinear system of equations. The following error function is
solved at each time step
Fi(X̂) = VLi (X̂) − VNL i(X̂)
and in vector form, the complete system is
 Pn
s
j=1 vL1,j (X̂, nt ) − vNL1 (X̂, nt )
 Pn s

j=1 vL2,j (X̂, nt ) − vNL2 (X̂, nt )
F(X̂) = 
 ..
 .
Pn s
j=1 vLns ,j (X̂, nt ) − vNLns (X̂, nt )
(6)






(7)
The error function in Eq. (7) is solved for each time step.
IV. Implementation
The formulation developed above resulted in a standard
nonlinear problem that can be solved using the Newton
method or quasi-Newton method. As the error function
CHRISTOFFERSEN et al.: STATE VARIABLE-BASED TRANSIENT ANALYSIS USING CONVOLUTION
changes only slightly from time step to time step, efficient
iterative matrix solve schemes can be used as a very good
preconditioner is available from the previous time step.
The general flow of the analysis is shown in Figure 1 and
was implemented in the Transim program.
3
shown in Fig. 2. The advantage of this topology is that
no extra nodes are added to the circuit and the size of the
MNAM is not changed. The augmentation network provides a better matrix conditioning for the I-FFT operation
at the expense of increased error due to finite numerical
accuracy. Ideally, the effect of the augmentation would be
exactly compensated. The resistive augmentation network
Parse Netlist
I Li
PREPROCESSING
VLi
VNLi
...
...
Create MNAM
I NLi
...
...
...
...
NONLINEAR
ELEMENT
Add Augmentation Network
Calculate Msv Matrix
...
LINEAR
SUBCIRCUIT
Do Filtering
...
...
I-FFT
NONLINEAR
ELEMENT
AT EACH TIME STEP
Convolution Sum
Solve Nonlinear System
AUGMENTATION
NETWORK
COMPENSATION
NETWORK
Iterate until convergence
Store Currents
Output Routines
Fig. 1. Flow diagram of the analysis code
A. Modified Nodal Admittance Matrix
Computation begins with the formulation and solution
of a frequency domain modified nodal admittance matrix
(MNAM) of the linear subcircuit. The MNAM routines
are based on the sparse matrix package Sparse 1.3 [17].
This is a flexible package of subroutines written in C that
quickly and accurately solve large sparse systems of linear equations. It also provides utilities such as MNAM
reordering and other utilities suited to circuit analysis. At
each frequency, the state variable impedance matrix MSV
is calculated from the LU decomposed MNAM [16].
B. Impulse Response Determination
The impulse response is obtained using the inverse real
Fourier transformation of each element of MSV . The transform requires special characteristics of the frequency domain variables at high frequencies so that aliasing is minimized. Problems occur with ideal inductors and capacitors
as the matrix elements can go to infinity. This can be corrected by cascading a resistive augmentation circuit with
the linear circuit and so ensure finite parameters [18], [1].
The use of scattering parameters achieves similar resistive
augmentation [7]. Being resistive, the effect of the augmentation network can be removed in transient simulation as
the resistors are unaffected by the Fourier transformation.
The current work uses the resistive augmentation network
LINEAR / NONLINEAR INTERFACE OF
AUGMENTED CIRCUIT
Fig. 2. Augmentation and compensation network
also serves to limit the extent of the impulse response as
energy is absorbed and multiple reflections are damped.
Again, note this effect is fully compensated.
Before converting the frequency domain MSV (impedancelike) matrix to the time domain, the imaginary part of the
frequency response needs to be band-limited so that the
function has a periodic (or circular) frequency response.
This significantly reduces the aliasing errors in the I-FFT
operation.
In Transim, frequency response limiting is implemented
by filtering the last part of the frequency spectrum of each
matrix element so the imaginary part goes to zero and the
real part goes to the DC value. In this way, the frequency
response is periodic. This is in contrast to low pass filtering [1] which zeroes the high frequency response and
introducing additional time-delay to take the imaginary response to zero [2]. The filtering factor ki is,
2
n−i
ki =
(8)
n − i0
where n is the number of discrete frequencies, i is the frequency index and i0 is the frequency index at which filtering starts. For all i greater than i0 , the filtering operation
on each state variable impedance matrix entry is
<{mi } = ki × <{mi } + (1 − ki ) × m0
={mi } = ki × ={mi }
(9)
(10)
The amount of filtering is an analysis option since in some
cases it is not necessary. By default, only the last 3% of
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
the spectrum is filtered to achieve an acceptable alias reduction. The effect on causality remains to be determined.
If this is not done the correct impulse response is not assured as the continuity condition is inherent in the I-FFT
(or FFT) operation.
Causality imposes special considerations for the response
obtained from the I-FFT operation. A discontinuity of the
impulse response generally occurs at t = 0, as the negative time response must be zero [8]. In this situation the
I-FFT operation yields a value which is the average of the
one-sided limits of the function at that point. That is the
value calculated at t = 0 is one half of the value at t = 0+ .
However, the upper side limit (at t = 0+ ) is required in
numerical integration and so the value of the I-FFT calculated time response must be multiplied by 2 to obtain the
required impulse response. Note that in Equation (5) the
first point of the impulse is divided by two, so actually the
multiplication can be saved.
Fourier transformation is implemented using a real IFFT algorithm from the FFTW library [19]. This algorithm requires only the positive frequency samples since the
negative frequency samples are the complex conjugate of
the corresponding positive frequencies. The FFTW library
is a comprehensive collection of C routines for computing
the discrete Fourier transform in one or more dimensions,
of both real and complex data, and of arbitrary input size.
C. Parameterized Nonlinear Models
Let the nonlinear subnetwork be described by the following generalized parametric equations [20]
vNL (t)
iNL (t)
dx
dn x
, . . . , n , xD (t)]
dt
dt
dx
dn x
= w[x(t),
, . . . , n , xD (t)]
dt
dt
= u[x(t),

x(t)



if x(t) ≤ V1
(14)
1
V
+
ln(1
+
α(x(t)
− V1 ))

1

α

if x(t) > V1

Is (exp(αx(t)) − 1)



if x(t) ≤ V1
=
(15)
I
exp(αV
)(1
+
α(x(t)
− V1 )) − Is

s
1


if x(t) > V1
v(t)
=
i(t)
where V1 is some threshold value. The model requires the
current, voltage and derivatives to be continuous at x = V1 .
This implies that
V1 =
ln(G1 /αIs )
α
(16)
where G is the slope ∂i/∂v and is chosen to be 5e8 · Is .
Note that using this value, V1 becomes independent of the
saturation current. In this way, the maximum value of the
exponential function is limited to 5e8.
The plots of Figures 3, 4 and 5 show the improvement
in the behavior of the model when using the state variable
approach. In a diode the current has an exponential dependence on voltage. This causes convergence problems when
the voltage is updated during nonlinear iteration. At voltages greater than the threshold, small voltage increments
can result in large current changes and hence changes in
the error function. The possibility of large changes is eliminated through the use of parameterization which ensures
smooth, well behaved current, voltage and error function
variations when the state variable is updated. See Figures 4
and 5.
(11)
0.6
(12)
0.5
where vNL (t), iNL (t) are vectors of voltages and currents
at the common ports, x(t) is a vector of state variables and
xD (t) a vector of time-delayed state variables, i.e., xDi(t) =
xi (t − τi). The time delays τi may be functions of the state
variables. All vectors in (12) have the same size nd equal
to the number of common (device) ports. This kind of
representation is convenient from the physical viewpoint,
as it is equivalent to a set of implicit integro-differential
equations in the port currents and voltages. This allows an
effective minimization of the number of subnetwork ports,
and what is more important, results in extreme generality
in device modeling capabilities.
Here a parametric diode model for the diode element is
presented [20]. The full model includes capacitances and
nonlinear resistance, but here a simple case is shown for
didactic purposes. The conventional current equation for
the diode is
i(t) = Is (exp(αv(t)) − 1)
as follows
(13)
and, based on previous work [20], the parametric model is
0.4
0.3
I (A)
4
0.2
0.1
0
−0.1
−1
−0.8
−0.6
−0.4
−0.2
0
V (V)
0.2
0.4
0.6
0.8
1
Fig. 3. Relation between v and i in a diode.
D. Convolution
Convolution of the impulse response of the linear circuit
with the outputs of the nonlinear elements has been proposed for transient analysis of distributed circuits with [7],
CHRISTOFFERSEN et al.: STATE VARIABLE-BASED TRANSIENT ANALYSIS USING CONVOLUTION
0.6
5
vides Newton and quasi-Newton methods and many options, such as the use of analytic Jacobian or forward,
backwards or central differences to approximate it, different quasi-Newton Jacobian updates and two globally convergent methods. This flexibility permits rapid simulator
development. Additionally, it reflects state-of-the-art numerical analysis.
0.5
0.4
I (A)
0.3
V. Simulation Of A Nonlinear Transmission Line
0.2
0.1
0
−0.1
−1
−0.5
0
0.5
1
1.5
2
2.5
X
Fig. 4. Relation between x and i in a diode.
1
0.8
0.6
0.4
V (V)
0.2
0
−0.2
−0.4
Nonlinear transmission lines (NLTLs) find applications
in a variety of high speed, wide bandwidth systems including picosecond resolution sampling circuits, laser and
switching diode drivers, test waveform generators, and mmwave sources [23]. They have three fundamental characteristics: nonlinearity, dispersion and dissipation. The NLTL
consist of coplanar waveguides (CPWs) periodically loaded
with reverse biased Schottky diodes. Diode-based NLTL
used for pulse generation are extremely nonlinear circuits
and are being used to test the robustness of circuit simulators. The NLTL considered here was designed with a
balance between the nonlinearity of the loaded nonlinear
elements and the dispersion of the periodic structure which
results in the formation of a stable soliton [24], [25]. The
nonlinearity of NLTLs is principally due to the voltage dependent capacitance of the diodes and the dissipation is
due to the conductor losses in the CPWs.
In this work, the NLTL was modeled using generic transmission lines with frequency-dependent loss and Schottky
diodes [26]. Skin effect was taken into account in the modeling of the transmission lines. The NLTL model is shown
in Figure 6 and is excited by a 9 GHz sinusoid. The NLTL
−0.6
50Ω
+
−0.8
−1
−1
−0.5
0
0.5
1
1.5
2
0011
0011
...
0011
50Ω
2.5
X
Fig. 5. Relation between x and v in a diode.
[2], [3], [4], [6], [1], [21]. The major problem identified with
this type of analysis is the rapid growth in computation and
memory requirements. The convolution integral, which becomes a convolution sum in computer implementations, is
O(n2M AX nS 2 ) where nM AX is the length of the impulse
response and nS is the number of state variables. Memory usage is O(nM AX × nS 2 ), principally due to storage
of the matrix impulse response. Thus reducing the number of state variables, nS , (i.e., using the minimum number
of state variables rather than the voltages at all nodes of
the nonlinear network) dramatically reduces memory and
compute time. As state variables permit the use of parameterized device models, the stability of the convolution
analysis is improved allowing larger time steps thus reducing nM AX .
E. Nonlinear Equation Solver
The nonlinear system at each time step is solved using
the NNES library [22]. It is written in Fortran and pro-
Fig. 6. Model of the nonlinear transmission line.
was designed for a 24 GHz initial Bragg frequency, 225 GHz
final Bragg frequency, 0.952097 tapering rule, and 120 ps
total compression. It contains 48 sections of CPW transmission lines and 47 diodes. The drive is a 27 dBm sine
wave with −3 V dc bias.
VI. Results and Discussion
Fig. 7 shows the calculated transient response of the soliton line including frequency dependent (skin effect) loss of
the transmission lines. The simulation time was 8 hours on
an UltraSPARC 1 workstation, clocked at 143 MHz. The
number of frequencies used to obtain the impulse response
was 4096 and the maximum frequency was 2700 GHz,
which corresponds to a time step of 0.185ps. The value
of the compensation resistor was 140 Ω. The comparison
among the measured output voltage across the load [23]
and transient and harmonic balance simulation (also using
Transim) is shown in Fig. 8. The harmonic balance analysis used the state variable approach described in [16]. The
convergence rate was increased and the number of harmonics reduced using the filtering technique described in [27],
6
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
30
0
25
−2
20
15
OUTPUT POWER (dBm)
VOLTAGE (V)
−4
−6
−8
−10
10
5
0
−5
−10
−12
−15
−14
0
100
200
300
400
500
TIME (ps)
600
700
800
900
−20
1000
0
50
100
150
200
SPECTRUM (GHz)
250
300
350
Fig. 7. Complete transient response for the soliton line
Fig. 9. Magnitude of the harmonics of the output power.
starting at harmonic number 30. The total number of frequencies was 40, and the simulation time was 30 hours on
the same computer. The harmonic content of the output
is shown in Fig. 9. The prediction of the main soliton am0
continues on identify the required time step for a prescribed
accuracy. The importance of IRC analysis of distributed
microwave circuits as opposed to conventional SPICE analysis is that frequency dependent linear elements can be
handled. In Fig. 11 the effect of frequency dependent trans-
−5
−2
−4
−10
VOLTAGE (V)
OUTPUT VOLTAGE (V)
0
EXPERIMENTAL DATA
HARMONIC BALANCE
TRANSIENT
−6
−8
−10
−15
0
10
20
30
TIME (ps)
40
50
60
−12
.18ps
.13ps
.27ps
Fig. 8. Comparison between experimental data and simulations
−14
380
plitude is correct, although the impulse width is slightly
smaller in the simulations. Note that the secondary soliton predicted by the simulations is not observed in the
measurements. This is probably due to neglecting higher
order parasitic effects of the interconnections. In both simulations, a parasitic inductance of 21.8 pH modeled the
connection between each diode and the CPW line. Without adding complexity this inductor was incorporated in
the nonlinear diode model. This is only possible with state
variables and results in a better conditioned transient analysis.
The transient analysis gives slightly different responses
depending on the size of the time step as shown in Fig. 10.
Smaller time step yields a simulation more accurate. Work
400
420
440
TIME (ps)
460
480
500
Fig. 10. Comparison between simulations using different time steps
mission line attenuation, principally due to the skin effect,
is compared to frequency-independent attenuation and no
attenuation. The frequency independent attenuation of the
transmission lines is the attenuation at 10 GHz. Frequency
dependent attenuation has only a small effect on the depth
and width of the soliton. The importance of modeling is
borne out by these results. Rather than focusing on the
previously deleterious effect of frequency dependent loss,
better modeling of the parasitics of the NLTL is required
to provide a better basis for computer aided design.
CHRISTOFFERSEN et al.: STATE VARIABLE-BASED TRANSIENT ANALYSIS USING CONVOLUTION
0
[7]
−2
[8]
VOLTAGE (V)
−4
−6
[9]
−8
[10]
−10
[11]
−12
ACTUAL
FREQUENCY−INDEPENDENT
NONE
−14
830
840
850
860
870
880
TIME (ps)
890
900
910
920
Fig. 11. Comparison between simulations using different attenuation
factors for the transmission lines
[12]
[13]
[14]
VII. Conclusion
The importance of the work presented here was the
development of a state-variable formulation for transient
analysis of distributed circuits using the impulse response
of the linear subnetwork and convolution. State variable based analysis minimizes compute time and memory requirements when the number of linear elements with
frequency-dependent loses is greater or about the same as
the number of nonlinear elements, but most importantly
it improves the robustness of the simulation by allowing
parameterized nonlinear device models to be used. The
transient simulation technique developed here is targeted
at the analysis of circuits with tightly coupled circuit and
field interactions. Modeling of the electromagnetic environment results in port descriptions without a defined global
reference node rendering the use of nodal voltages is problematic [28], [29]. The current implementation of Transim circumvents this problem by the use of local reference
nodes.
[15]
[16]
[17]
[18]
[19]
[20]
[21]
References
[1]
[2]
[3]
[4]
[5]
[6]
M. S. Basel, M. B. Steer and P. D. Franzon, “Simulation of
high speed interconnects using a convolution-based hierarchical
packaging simulator,” IEEE Trans. on Components, Packaging,
and Manufacturing Techn., Vol. 18, February 1995, pp. 74-82.
T. J. Brazil, “A new method for the transient simulation of
causal linear systems described in the frequency domain,” 1992
IEEE MTT-S Int. Microwave Symp. Digest, June 1992, pp.
1485-1488.
P. Perry and T. J. Brazil, “Hilbert-transform-derived relative
group delay,” IEEE Trans. on Microwave Theory and Techn.,
Vol 45, Aug. 1997, pt. 1, pp. 1214-1225.
T. J. Brazil, “Causal convolution—a new method for the transient analysis of linear systems at microwave frequencies,” IEEE
Trans. on Microwave Theory and Techn., Vol. 43, Feb. 1995, pp.
315-23.
A. R. Djordjevic and T. K. Sarkar, “Analysis of time response of
lossy multiconductor transmission line networks,” IEEE Trans.
on Microwave Theory and Techn., Vol. MTT-35, Oct. 1987, pp.
898-908.
D. Winkelstein, R. Pomerleau and M. B. Steer, “Transient simulation of complex, lossy, multi-port transmission line networks
with nonlinear digital device termination using a circuit simu-
[22]
[23]
[24]
[25]
[26]
[27]
7
lator,” Conf. Proc. IEEE SOUTHEASTCON, Vol. 3, pp. 12391244.
J. E. Schutt-Aine and R. Mittra, “Nonlinear transient analysis
of coupled transmission lines,” IEEE Trans. on Circuits and
Systems, Vol. 36, Jul. 1989, pp. 959-967.
P. Stenius, P. Heikkilä and M. Valtonen, “Transient analysis of
circuits including frequency-dependent components using transgyrator and convolution,” Proc. of the 11th European Conference on Circuit Theory and Design, Part II, 1993, pp. 1299-1304.
P. K. Chan, Comments on “Asymptotic waveform evaluation for
timing analysis,” IEEE Trans. on Computer Aided Design, Vol.
10, Aug. 1991, pp. 1078-79.
R. Archar, M. S. Nakhla and Qi-Jun Zhang, “Full-wave analysis
of high speed interconnects using complex frequency hoping,”
IEEE Trans. on Computer-Aided Design of Integrated Circuits
and Systems, Oct. 1998, pp 997-1010.
M. Celik, O. Ocali, M. A. Tan, and A. Atalar, “Pole-zero computation in microwave circuits using multipoint Padé approximation,” IEEE Trans. on Circuits and Systems, Jan. 1995, pp.
6-13.
E. Chiprout and M. Nakhla, “Fast nonlinear waveform estimation for large distributed networks,” 1992 IEEE MTT-S Int.
Microwave Symp. Digest, Vol.3, Jun. 1992, pp. 1341-1344.
R. J. Trihy and Ronald A. Rohrer, “AWE macromodels for nonlinear circuits,” Proceedings of the 36th Midwest Symposium on
Circuits and Systems, Vol. 1, Aug. 1993, pp. 633-636.
R. Griffith and M. S. Nakhla, “Mixed frequency/time domain
analysis of nonlinear circuits,” IEEE Trans. on Computer Aided
Design, Vol.11, Aug. 1992, pp. 1032-43.
M. S. Nakhla and J. Vlach, “A piecewise harmonic balance technique for determination of periodic response of nonlinear systems,” IEEE Trans. Circuits Systems, Vol. 23, Feb. 1976, pp.
85-91.
C. E. Christoffersen, M. B. Steer and M. A. Summers, “Harmonic balance analysis for systems with circuit-field interactions,” 1998 IEEE Int. Microwave Symp. Digest, June 1998,
pp. 1131-1134.
K. S. Kundert and A. Songiovanni-Vincentelli, Sparse user’s
guide - a sparse linear equation solver, Dept. of Electrical Engineering and Computer Sciences, University of California, Berkeley, Calif. 94720, Version 1.3a, Apr 1988.
M. Ozkar, Transient analysis of spatially distributed microwave
circuits using convolution and state variables, M. S. Thesis, Department of Electrical and Computer Engineering, North Carolina State University.
M. Frigo and S. G. Johnson, FFTW User’s Manual, Massachusetts Institute of Technology, September 1998.
V. Rizzoli, A. Lipparini, A. Costanzo, F. Mastri, C. Ceccetti, A.
Neri and D. Masotti, “State-of-the-art harmonic-balance simulation of forced nonlinear microwave circuits by the piecewise
technique,” IEEE Trans. on Microwave Theory and Techn., Vol.
40, Jan 1992, pp 12-27.
C. Gordon, T. Blazeck and R. Mittra, “Time domain simulation
of multiconductor transmission lines with frequency-dependent
losses,” IEEE Trans. on Computer Aided Design of Integrated
Circuits and Systems, Vol. 11 Nov. 1992 pp. 1372-87.
R. S. Bain, NNES user’s manual, 1993.
M. G. Case, Nonlinear transmission lines for picosecond pulse,
impulse and millimeter-wave harmonic generation, Ph.D Dissertation, Department of Electrical and Computer Engineering, University of California, Santa Barbara, California, U.S.A.,
1993.
M. J. W. Rodwell, M. Kamegawa, R. Yu, M. Case, E. Carman
and K. S. Giboney, “GaAs nonlinear transmission lines for picosecond pulse generation and millimeter-wave sampling,” IEEE
Trans. on Microwave Theory and Techn., Vol. 39, July 1991, pp.
1194-1204.
H. Shi, C. W. Domier, N. C. Luhmann, “A monolithic nonlinear
transmission line system for the experimental study of lattice
solutions,” J. of Applied Physics, Vol 4., August 1995, pp. 255864.
Compact Software, Microwave Harmonica Elements Library,
1994.
A. Brambilla and D. D’Amore, “A filter-based technique for
the harmonic balance method,” IEEE Trans. on Circuits and
Systems—I: Fundamental Theory and Applications, Vol. 43,
Feb. 1996, pp. 92-98.
8
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
[28] A. I. Khalil and M. B. Steer “Circuit theory for spatially distributed microwave circuits,” IEEE Trans. on Microwave Theory and Techn., Vol. 46, Oct. 1998, pp 1500-1503.
[29] C. E. Christoffersen and M. B. Steer “Implementation of the
local reference concept for spatially distributed circuits,” Int. J.
of RF and Microwave Computer-Aided Eng., In Press.
Carlos E. Christoffersen (S’91) was born
in Argentina in 1968. He received the Electronic Engineer degree at the National University of Rosario, Argentina in 1993. From 1993
to 1995 he was research fellow of the National
Research Council of Argentina (CONICET).
He received the M.S. degree in electrical engineering in 1998 from North Carolina State
University. Currently he is pursuing his Ph.D.
degree at the same university. Since 1996, he
is with the Department of Electrical and Computer Engineering as a research assistant. His research interests include computer aided analysis of circuits, analog, RF and microwave
circuit design.
Mete Ozkar (S’97) received the B.S. degree in electrical engineering from Middle East
Technical University, Ankara, Turkey, in 1996,
and the M.S. degree from North Carolina State
University, Raleigh, North Carolina, in 1998.
He is currently working toward the Ph.D degree in electrical engineering and is a Research
Assistant with the Electronics Research Laboratory, Department of Electrical and Computer
engineering, North Carolina State University,
where he performs research in the simulation
and the development of spatial power combining systems.
Michael Case (S’87-M’93-SM’98) was born in
1966 in Ventura, California. After receiving his
B. S. degree in 1989 from U. C. Santa Barbara, he began researching there under a state
fellowship for Professor Mark Rodwell. In Dr.
Rodwell’s group he studied nonlinear transmission lines for applications in high-speed waveform shaping and signal detection. Michael
earned his M. S. and Ph. D. degrees in 1991
and 1993 respectively from U. C. Santa Barbara. The Air Force Office of Scientific Research sponsored a majority of his work. HRL Laboratories (previously the Hughes Aircraft Company’s research labs) has employed
Michael in Malibu, California since 1993. He is currently involved
with millimeter-wave device characterization, circuit design, and measurement techniques.
Mark Rodwell was born in Altrincham, England in 1960. He received the B.S. degree in
electrical engineering from the University of
Tennessee, Knoxville, in 1980, and the M.S.
(1982) and Ph.D. degrees (1988) in electrical
engineering from Stanford University. From
1982 through 1984 he worked at AT&T Bell
Laboratories, developing optical transmission
systems. He was a research associate at Stanford University from January to September
1988.
In September 1988 he joined the Department of Electrical and
Computer Engineering, at the University of California, Santa Barbara, where he is currently Professor and Director of the Compound Semiconductor Research Laboratories. His current research
involves submicron scaling of millimeter-wave heterojunction bipolar transistors (HBTs) and development of HBT integrated circuits
for microwave mixed-signal ICs and fiber optic transmission systems. His group has developed deep submicron Schottky-collector
resonant-tunnel diodes with THz bandwidths, and has developed
monolithic submillimeter-wave oscillators with these devices. His
group has worked extensively in the area of GaAs Schottky-diode integrated circuits for subpicosecond pulse generation, signal sampling
at submillimeter-wave bandwidths, and millimeter-wave instrumentation. He is the recipient of a 1989 National Science Foundation Presidential Young Investigator award, and his work on submillimeterwave diode ICs was awarded the 1997 IEEE Microwave Prize .
Michael B. Steer (S’76—M’78—SM’90—F’99) for a photograph and biography, see p. 13 of the January 1999 issue
of this Transactions.